1. Numerical Model Atmospheres (Hubeny & Mihalas 16, 17)
Equations Hydrostatic Equilibrium Temperature Correction Schemes

2. Summary: Basic Equations
Corresponding State Parameter
Radiative transfer
Mean intensities, Jν
Radiative equilibrium
Temperature, T
Hydrostatic equilibrium
Total particle density, N
Statistical equilibrium
Populations, ni
Charge conservation
Electron density, ne

3. Physical State
Recall rate equations that link the populations in each ionization/excitation state
Based primarily upon temperature and electron density
Given abundances, ne, T we can find N, Pg, and ρ
With these state variables, we can calculate the gas opacity as a function of frequency

4. Hydrostatic Equilibrium
Gravitational force inward is balanced by the pressure gradient outwards,
Pressure may have several components: gas, radiation, turbulence, magnetic
μ = # atomic mass units / free particle in gas

5. Column Density
Rewrite H.E. using column mass inwards (measured in g/cm2), “RHOX” in ATLAS
Solution for constant T, μ (scale height):

6. Gas Pressure Gradient Ignoring turbulence and magnetic fields:
Radiation pressure acts against gravity (important in O-stars, supergiants)

7. Temperature Relations
If we knew T(m) and P(m) then we could get ρ(m) (gas law) and then find χν and ην
Then solve the transfer equation for the radiative field (Sν = ην / χν )
But normally we start with T(τ) not T(m)
Since dm = -ρ dz = dτν / κν we can transform results to an optical depth scale by considering the opacity

8. ATLAS Approach (Kurucz)
H.E.
Start at top and estimate opacity κ from adopted gas pressure and temperature
At next optical depth step down,
Recalculate κ for mean between optical depth steps, then iterate to convergence
Move down to next depth point and repeat

9. Temperature Distributions
If we have a good T(τ) relation, then model is complete: T(τ) → P(τ) → ρ(τ) → radiation field
However, usually first guess for T(τ) will not satisfy flux conservation at every depth point
Use temperature correction schemes based upon radiative equilibrium

10. Solar Temperature Relation
From Eddington-Barbier (limb darkening)
τ0 = τ(5000 Å)

11. Rescaling for Other Stars
Reasonable starting approximation

12. Temperature Relations for Supergiants
Differences small despite very different length scales

13. Including line opacity or line blanketing
Other Effects on T(τ)
Including line opacity or line blanketing
Convection

14. Temperature Correction Schemes
“The temperature correction need not be very accurate, because successive iterations of the model remove small errors. It should be emphasized that the criterion for judging the effectiveness of a temperature correction scheme is the total amount of computer time needed to calculate a model. Mathematical rigor is irrelevant. Any empirically derived tricks for speeding convergence are completely justified.” (R. L. Kurucz)

15. Some T Correction Methods
Λ iteration scheme
Not too good at depth (cf. gray case)

16. Some T Correction Methods
Unsöld-Lucy method similar to gray case: find corrections to the source function = Planck function that keep flux conserved (good for LTE, not non-LTE)
Avrett and Krook method (ATLAS) develop perturbation equations for both T and τ at discrete points (important for upper and lower depths, respectively); interpolate back to standard τ grid at end (useful even when convection carries a significant fraction of flux)

17. Some T Correction Methods
Auer & Mihalas (1969, ApJ, 158, 641) linearization method: build in ΔT correction in Feautrier method
Matrices more complicated
Solve for intensities then update ΔT